Mathematics for the interested outsider

The Hodge Star in Coordinates

It will be useful to be able to write down the Hodge star in a local coordinate system. So let’s say that we’re in an oriented coordinate patch of an oriented Riemannian manifold , which means that we have a canonical volume form that locally looks like

Now, we know that any -form on can be written out as a sum of functions times -fold wedges:

Since the star operation is linear, we just need to figure out what its value is on the -fold wedges. And for these the key condition is that for every -form we have

Since both sides of this condition are linear in , we also only need to consider values of which are -fold wedges. If is not the same wedge as , then the inner product is zero, while if then

And so must be times the -fold wedge made up of all the that do not show up in . The positive or negative sign is decided by which order gives us an even permutation of all the on the left-hand side of the above equation.

Hi John, just an observation. When you pick local coordinates, the frame for the cotangent bundle dx_{i} i=1,,,,,n induced by the local coordinates is not orthonormal in general, so the inner product of two different wedges is not zero. In the same way is not delta_{i,j}. I think this only works if you pick Riemannian normal coordinates centered at some point “p”, even in this case this expression for the Hodge star operator would be valid only for the center point “p” and not for other points in this chart.

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